Theorem pwpwuni 45894

Description: Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
pwpwuni	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))

Proof of Theorem pwpwuni

Step	Hyp	Ref	Expression
1		elpwg 4560	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝐴 ⊆ 𝒫 𝐵))
2		sspwuni 5060	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
4		uniexg 7743	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
5		elpwg 4560	. . . 4 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
6	4, 5	syl 18	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
7	6	bicomd 226	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ⊆ 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))
8	1, 3, 7	3bitrd 308	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2145  Vcvv 3450   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-ss 3916  df-pw 4559  df-uni 4868

This theorem is used by:  psmeasurelem  47301